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Mirrors  >  Home  >  MPE Home  >  Th. List  >  carden2b Structured version   Visualization version   GIF version
Theorem carden2b 9958
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9957 are meant to replace carden 10542 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Proof of Theorem carden2b
StepHypRef Expression
1 cardne 9956 . . . . 5 ((cardβ€˜π΅) ∈ (cardβ€˜π΄) β†’ Β¬ (cardβ€˜π΅) β‰ˆ 𝐴)
2 ennum 9938 . . . . . . . 8 (𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ dom card ↔ 𝐡 ∈ dom card))
32biimpa 477 . . . . . . 7 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 ∈ dom card)
4 cardid2 9944 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
53, 4syl 17 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
6 ensym 8995 . . . . . . 7 (𝐴 β‰ˆ 𝐡 β†’ 𝐡 β‰ˆ 𝐴)
76adantr 481 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 β‰ˆ 𝐴)
8 entr 8998 . . . . . 6 (((cardβ€˜π΅) β‰ˆ 𝐡 ∧ 𝐡 β‰ˆ 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
95, 7, 8syl2anc 584 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
11 cardon 9935 . . . . 5 (cardβ€˜π΄) ∈ On
12 cardon 9935 . . . . 5 (cardβ€˜π΅) ∈ On
13 ontri1 6395 . . . . 5 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄)))
1411, 12, 13mp2an 690 . . . 4 ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
1510, 14sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
16 cardne 9956 . . . . 5 ((cardβ€˜π΄) ∈ (cardβ€˜π΅) β†’ Β¬ (cardβ€˜π΄) β‰ˆ 𝐡)
17 cardid2 9944 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
18 id 22 . . . . . 6 (𝐴 β‰ˆ 𝐡 β†’ 𝐴 β‰ˆ 𝐡)
19 entr 8998 . . . . . 6 (((cardβ€˜π΄) β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ 𝐡) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2017, 18, 19syl2anr 597 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2116, 20nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
22 ontri1 6395 . . . . 5 (((cardβ€˜π΅) ∈ On ∧ (cardβ€˜π΄) ∈ On) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅)))
2312, 11, 22mp2an 690 . . . 4 ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
2421, 23sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) βŠ† (cardβ€˜π΄))
2515, 24eqssd 3998 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
26 ndmfv 6923 . . . 4 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
2726adantl 482 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = βˆ…)
282notbid 317 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (Β¬ 𝐴 ∈ dom card ↔ Β¬ 𝐡 ∈ dom card))
2928biimpa 477 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ Β¬ 𝐡 ∈ dom card)
30 ndmfv 6923 . . . 4 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
3129, 30syl 17 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) = βˆ…)
3227, 31eqtr4d 2775 . 2 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
3325, 32pm2.61dan 811 1 (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  dom cdm 5675  Oncon0 6361  β€˜cfv 6540   β‰ˆ cen 8932  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-card 9930
This theorem is referenced by:  card1  9959  carddom2  9968  cardennn  9974  cardsucinf  9975  pm54.43lem  9991  nnadju  10188  nnadjuALT  10189  ficardun  10191  ficardunOLD  10192  ackbij1lem5  10215  ackbij1lem8  10218  ackbij1lem9  10219  ackbij2lem2  10231  carden  10542  r1tskina  10773  cardfz  13931
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